Metropolis algotithm works when the proba of proposing A from B is the same as the proba of proposing B from A. The proposal distribution is symmetric while it not for Gibbs sampling. With Gibbs sampling, you get a good estimate of the posterior with many fewer samples than Metropolis approach. Gibbs sampling computes these adaptive proposals using particular combinations of prior distributions and likelihoods known as conjugate pairs. Metropolis and Gibbs can get stuck in a narrow region of parameter space.

Hamiltonian Monte Carlo is more efficient for proposals but computationally costly. It needs less samples so less computer time. Autocorrelation is pretty low. The warmup phase try to figure out which step size eplore the posterior efficiently. During the warmup, no samples are produce (while the bun-in phase in Gibbs produce samples). The no-U-TURM SAMPLER (or NUTS) figures out the number of leapfrog steps. The NUTS uses the shape of the posterior to infer when the path is turning around.

Remember that some posterior distributions are very difficult to sample from, for any algorithms.

Practice

E6

Criteria for healthy Markov chains:
1) stationarity
2) good mixing
3) convergence

dat_slim <- list(
    log_gdp_std = dd$log_gdp_std,
    rugged_std = dd$rugged_std,
    cid = as.integer( dd$cid )
)
str(dat_slim)
## List of 3
##  $ log_gdp_std: num [1:170] 0.88 0.965 1.166 1.104 0.915 ...
##  $ rugged_std : num [1:170] 0.138 0.553 0.124 0.125 0.433 ...
##  $ cid        : int [1:170] 1 2 2 2 2 2 2 2 2 1 ...
m9.1 <- ulam(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.1 ) ,
        b[cid] ~ dnorm( 0 , 0.3 ) ,
        sigma ~ dexp( 1 )
    ) , data=dat_slim , chains=4 , cores=4 )
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Users/Anabelle_Laurent/Library/R/3.5/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/unsupported"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/BH/include" -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/src/"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -D_REENTRANT  -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
traceplot(m9.1)

nonhealthy <- ulam(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.0001 ) ,
        b[cid] ~ dnorm( 0 , 0.0003 ) ,
        sigma ~ dexp( 1 )
    ) , data=dat_slim , chains=4 , cores=4 )
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Users/Anabelle_Laurent/Library/R/3.5/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/unsupported"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/BH/include" -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/src/"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -D_REENTRANT  -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
traceplot(nonhealthy)

We can see that the trace for sigma is malfunctioning! Also n_eff is pretty small

E7

trankplot(m9.1)
trankplot(nonhealthy)

M1

modM1<- ulam(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.1 ) ,
        b[cid] ~ dnorm( 0 , 0.3 ) ,
        sigma ~ dunif(0,1)
    ) , data=dat_slim , chains=4 , cores=4 )
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Users/Anabelle_Laurent/Library/R/3.5/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/unsupported"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/BH/include" -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/src/"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -D_REENTRANT  -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
pairs(m9.1)

pairs(modM1)

The dunif prior for sigma has a undectible influence on the posterior distribution.

M.2

modM2<- ulam(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.1 ) ,
        b[cid] ~ dexp( 0.3 ) ,
        sigma ~ dunif(0,1)
    ) , data=dat_slim , chains=4 , cores=4 )
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Users/Anabelle_Laurent/Library/R/3.5/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/unsupported"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/BH/include" -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/src/"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -D_REENTRANT  -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
pairs(modM2)

posterior distribution of b.2 is positive. The dexp prior gives more importance to positive values than dnorm(0,0.3)

M.3

modM3 <- ulam(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.1 ) ,
        b[cid] ~ dnorm( 0 , 0.3 ) ,
        sigma ~ dexp( 1 )
    ) , data=dat_slim , chains=4 , cores=4 , iter=1000)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Users/Anabelle_Laurent/Library/R/3.5/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/unsupported"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/BH/include" -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/src/"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -D_REENTRANT  -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
precis(modM3,depth = 2)
##             mean          sd        5.5%       94.5%    n_eff     Rhat4
## a[1]   0.8863281 0.016122737  0.86016457  0.91162345 2239.271 0.9988868
## a[2]   1.0506759 0.010137093  1.03458888  1.06651262 2634.063 0.9996079
## b[1]   0.1299768 0.070541534  0.02099318  0.24834442 2165.037 1.0000979
## b[2]  -0.1425559 0.056383411 -0.23408657 -0.05394722 2439.010 1.0002718
## sigma  0.1113449 0.006366527  0.10148816  0.12163253 1903.962 1.0017080

It seems working with the default value (niter=1000). We have 4 chains so 2000 samples (because half is for warmup).

modM3bis <- ulam(
    alist(
        log_gdp_std ~ dnorm( mu , sigma ) ,
        mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
        a[cid] ~ dnorm( 1 , 0.1 ) ,
        b[cid] ~ dnorm( 0 , 0.3 ) ,
        sigma ~ dexp( 1 )
    ) , data=dat_slim , chains=4 , cores=4 , iter=500)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Users/Anabelle_Laurent/Library/R/3.5/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/unsupported"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/BH/include" -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/src/"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -D_REENTRANT  -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
precis(modM3bis,depth = 2)
##             mean          sd        5.5%       94.5%    n_eff     Rhat4
## a[1]   0.8859459 0.016347565  0.86017495  0.91180719 1344.075 1.0003122
## a[2]   1.0501916 0.009929244  1.03417127  1.06568342 1499.588 0.9978082
## b[1]   0.1328406 0.077808844  0.00604712  0.24967335 1052.691 1.0020480
## b[2]  -0.1427206 0.055307460 -0.23398687 -0.05651928 1279.944 1.0020324
## sigma  0.1117807 0.006136977  0.10258318  0.12180140 1183.639 0.9982488

It is not working when iter=500 as the effective sample size is too low (see error message). Rhat is still okay (see traceplot for convergence).

traceplot(modM3bis)

H1

mp <- ulam(
    alist(
      a ~ dnorm( 1 , 0.1 ) ,
      b ~ dcauchy( 0 ,1 )
    ), data=list(y=1),chains=1,iter=2000)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Users/Anabelle_Laurent/Library/R/3.5/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/unsupported"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/BH/include" -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/src/"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -D_REENTRANT  -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL 'ee52ba3dc53fc5ecd3df266f8e764a95' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 9e-06 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.09 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
## Chain 1: 
## Chain 1: Iteration:    1 / 2000 [  0%]  (Warmup)
## Chain 1: Iteration:  200 / 2000 [ 10%]  (Warmup)
## Chain 1: Iteration:  400 / 2000 [ 20%]  (Warmup)
## Chain 1: Iteration:  600 / 2000 [ 30%]  (Warmup)
## Chain 1: Iteration:  800 / 2000 [ 40%]  (Warmup)
## Chain 1: Iteration: 1000 / 2000 [ 50%]  (Warmup)
## Chain 1: Iteration: 1001 / 2000 [ 50%]  (Sampling)
## Chain 1: Iteration: 1200 / 2000 [ 60%]  (Sampling)
## Chain 1: Iteration: 1400 / 2000 [ 70%]  (Sampling)
## Chain 1: Iteration: 1600 / 2000 [ 80%]  (Sampling)
## Chain 1: Iteration: 1800 / 2000 [ 90%]  (Sampling)
## Chain 1: Iteration: 2000 / 2000 [100%]  (Sampling)
## Chain 1: 
## Chain 1:  Elapsed Time: 0.048317 seconds (Warm-up)
## Chain 1:                0.053569 seconds (Sampling)
## Chain 1:                0.101886 seconds (Total)
## Chain 1:
precis(mp)
##        mean         sd       5.5%    94.5%    n_eff     Rhat4
## a 0.9886442  0.1097502  0.8079348 1.155640 184.2202 0.9990673
## b 0.9024317 12.0270954 -6.5087773 5.807344 141.9182 1.0039142
traceplot(mp)

ESS was too low so I updated the model mp with iter=2000 (was still too low). As it has no expected value, we can see some spikes looking at the traceplot of parameter b.

H2

data(WaffleDivorce)
d <- WaffleDivorce
d$D <- standardize( d$Divorce )
d$M <- standardize( d$Marriage )
d$A <- standardize( d$MedianAgeMarriage )
sd( d$MedianAgeMarriage )
## [1] 1.24363
dat_d<- list(
    A = d$A,
    M= d$M,
    D=d$D
)
str(dat_d)
## List of 3
##  $ A: num [1:50] -0.606 -0.687 -0.204 -1.41 0.6 ...
##   ..- attr(*, "scaled:center")= num 26.1
##   ..- attr(*, "scaled:scale")= num 1.24
##  $ M: num [1:50] 0.0226 1.5498 0.049 1.6551 -0.267 ...
##   ..- attr(*, "scaled:center")= num 20.1
##   ..- attr(*, "scaled:scale")= num 3.8
##  $ D: num [1:50] 1.654 1.544 0.611 2.094 -0.927 ...
##   ..- attr(*, "scaled:center")= num 9.69
##   ..- attr(*, "scaled:scale")= num 1.82
m5.1 <- ulam(
    alist(
        D ~ dnorm( mu , sigma ) ,
        mu <- a + bA * A ,
        a ~ dnorm( 0 , 0.2 ) ,
        bA ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = dat_d , chains=3,log_lik=TRUE)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Users/Anabelle_Laurent/Library/R/3.5/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/unsupported"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/BH/include" -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/src/"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -D_REENTRANT  -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL 'bc675254cc2159bb33743bbf24468ab3' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 2.9e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.29 seconds.
## Chain 1: Adjust your expectations accordingly!
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## Chain 1: 
## 
## SAMPLING FOR MODEL 'bc675254cc2159bb33743bbf24468ab3' NOW (CHAIN 2).
## Chain 2: 
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## Chain 2: Adjust your expectations accordingly!
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## Chain 2: 
## 
## SAMPLING FOR MODEL 'bc675254cc2159bb33743bbf24468ab3' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 1e-05 seconds
## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.1 seconds.
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## Chain 3:
m5.2 <- ulam(
    alist(
       D ~ dnorm( mu , sigma ) ,
        mu <- a + bM * M ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = dat_d , chains=3,log_lik=TRUE)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Users/Anabelle_Laurent/Library/R/3.5/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/unsupported"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/BH/include" -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/src/"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -D_REENTRANT  -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '8b690dbf8986a1e80daeae2b8293f78f' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 2.2e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.22 seconds.
## Chain 1: Adjust your expectations accordingly!
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## Chain 1: 
## 
## SAMPLING FOR MODEL '8b690dbf8986a1e80daeae2b8293f78f' NOW (CHAIN 2).
## Chain 2: 
## Chain 2: Gradient evaluation took 1e-05 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.1 seconds.
## Chain 2: Adjust your expectations accordingly!
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## Chain 2: 
## 
## SAMPLING FOR MODEL '8b690dbf8986a1e80daeae2b8293f78f' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 8e-06 seconds
## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.08 seconds.
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## Chain 3:
m5.3 <- ulam(
    alist(
D ~ dnorm( mu , sigma ) ,
        mu <- a + bM*M + bA*A ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        bA ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = dat_d , chains=3,log_lik=TRUE)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Users/Anabelle_Laurent/Library/R/3.5/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/unsupported"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/BH/include" -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/src/"  -I"/Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/3.5/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -D_REENTRANT  -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Users/Anabelle_Laurent/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL 'd27a7128c546624a07b356f67a87a125' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 2.6e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.26 seconds.
## Chain 1: Adjust your expectations accordingly!
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## Chain 1: 
## 
## SAMPLING FOR MODEL 'd27a7128c546624a07b356f67a87a125' NOW (CHAIN 2).
## Chain 2: 
## Chain 2: Gradient evaluation took 1.8e-05 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.18 seconds.
## Chain 2: Adjust your expectations accordingly!
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## Chain 2: 
## 
## SAMPLING FOR MODEL 'd27a7128c546624a07b356f67a87a125' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 1e-05 seconds
## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.1 seconds.
## Chain 3: Adjust your expectations accordingly!
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## Chain 3:
compare(m5.1,m5.2,m5.3,func = WAIC)  
##          WAIC       SE     dWAIC       dSE    pWAIC       weight
## m5.1 125.7338 12.65334  0.000000        NA 3.642137 0.6813552207
## m5.3 127.2582 12.56555  1.524466 0.7194587 4.550860 0.3179361899
## m5.2 139.4709  9.74207 13.737125 9.2098881 3.006819 0.0007085895